Question

Suppose you make and sell skin lotion. A bottle of regular skin lotion contains 2 c oil and 1 c cocoa butter. A bottle of extra-rich skin lotion contains 1 c oil and 2 c cocoa butter. You will make a profit of $10/bottle on regular lotion and a profit of $8/bottle on extra-rich lotion. You have 24 c oil and 18 c cocoa butter.

a. How many bottles of each type of lotion should you make to maximize your profit? (2 points)
b. What is the maximum profit? ( 2 points)

Answer 1

  The analysis in previous answer is close, but it 
suffers from being wrong. 

The problem asks what you profit can be made from 
24 cups of oil and 18 cups of cocoa butter. 

The answer of 7 regular + 9 rich uses: 

    cocoa butter: 7*1+9*2 = 25 cups cocoa butter 
    oil : 7*2+9*1 = 23 cups oil 

So it violate *both* constraints. 



The profit formula is correct: P=10*r + 8*e 

and the constraints are: 

    2r + 1x <= 24 
    1r + 2x <= 18 

The previous analysis just went wrong when it included 
points which violated the constraints. The correct 
answer is 10 regular + 4 extra = $132 profit. 

    cocoa butter: 10*1+4*2 = 18 cups cocoa butter 
    oil: 10*2+4*1 = 24 cups of oil 

    Profit = 10*10 + 4*8 = $132